Finite Group Databasebeta

← #11713904

Subgroups — #11713904

Order 768.

462 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic
2#73148C211yesmaximal, characteristic
3#73148C211yesmaximal, characteristic
4C4⋊#859C211yesmaximal, characteristic, complemented
5#73059C211yesmaximal, characteristic, complemented
6#73059C211yesmaximal, characteristic, complemented
7C4×#859C211yesmaximal, characteristic, complemented
8#27522—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C2×#879—41nomaximal
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×#859C2211yescharacteristic
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17#2496—31no—
18Q8×C22⋊C4D311yescharacteristic
19#2077—31no—
20#2077—31no—
21#2072—31no—
22#2066—31no—
23#1933—31no—
24#1927—31no—
25#1923—31no—
26#1865—31no—
27#1865—31no—
28#1803—31no—
29#1787—31no—
30#1787—31no—
31C4×#103—31no—
32C22×SL(2,𝔽3)—21no—
33C22×SL(2,𝔽3)—21no—
34C22×SL(2,𝔽3)—21no—
35C22×SL(2,𝔽3)C2311yescharacteristic
36C2×C3×C22⋊C4—41no—
37C2×#217—41no—
38C2×#217—41no—
39C2×C4⋊Dic3—41no—
40C2×Dic3⋊C4—41no—
41C2×Dic3⋊C4—41no—
42C2×C4×Dic3—41no—
43#879—48no—
44C4×SL(2,𝔽3)—21no—
45C4×SL(2,𝔽3)—21no—
46#859—21no—
47#859—21no—
48#859—21no—
49#859—21no—
50C23×Q8C2×D311yescharacteristic
51C2×#126—61no—
52C2×C4×Q8C2×D311yescharacteristic
53C2×C4×Q8C2×D311yescharacteristic
54C2×#106—31no—
55C2×#106—31no—
56C2×C4⋊C8—31no—
57C2×#103—31no—
58C2×#103—31no—
59C2×#103—31no—
60C2×#103—31no—
61C2×#103—31no—
62C2×#103—31no—
63C2×#103—31no—
64C2×#103—31no—
65C2×C22⋊C8—31no—
66C2×C22⋊C8—31no—
67C2×C4×C8—31no—
68#400—61no—
69#397—61no—
70#391—31no—
71#391—31no—
72#389—31no—
73#389—31no—
74#387—61no—
75#384—31no—
76#380—31no—
77#380—31no—
78#380—31no—
79C4×C4⋊C4—31no—
80C4×C22⋊C4—31no—
81#340—31no—
82#340—31no—
83#340—31no—
84#340—31no—
85#340—31no—
86#340—31no—
87#336—31no—
88#336—31no—
89C24×C3—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C2×SL(2,𝔽3)—41no—
97C2×SL(2,𝔽3)#4111yescharacteristic
98C2×SL(2,𝔽3)#4111yescharacteristic
99C2×SL(2,𝔽3)#4111yescharacteristic
100C3×C22⋊C4—44no—
101#217—44no—
102#217—44no—
103C4⋊Dic3—44no—
104Dic3⋊C4—44no—
105Dic3⋊C4—44no—
106C4×Dic3—44no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—31no—
111C22×Q8—61no—
112C22×Q8—61no—
113C22×Q8—61no—
114C22×Q8C22×D311yescharacteristic
115C23×C4—31no—
116C22×C8—31no—
117C22×C8—31no—
118C22×C8—31no—
119C22×C8—31no—
120#126—124no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8—61no—
124C4×Q8—61no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4S411yescharacteristic
139C2×C42—31no—
140C2×C42—31no—
141C2×C42—61no—
142#106—121no—
143#106—121no—
144C4⋊C8—121no—
145#103—61no—
146#103—61no—
147#103—61no—
148#103—61no—
149#103—62no—
150#103—62no—
151#103—61no—
152#103—61no—
153#103—61no—
154#103—61no—
155#103—121no—
156#103—121no—
157C22⋊C8—121no—
158C22⋊C8—121no—
159C4×C8—121no—
160#95—31no—
161#95—31no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168C23×C3—42no—
169C23×C3—41no—
170C23×C3—81no—
171C23×C3—81no—
172C23×C3—81no—
173C23×C3—81no—
174C23×C3—81no—
175C23×C3—81no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—81no—
179C2×C3×C4—81no—
180C2×C3×C4—81no—
181C2×C3×C4—81no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—81no—
187C2×Dic3—81no—
188C2×Dic3—81no—
189C2×Dic3—81no—
190C2×Dic3—81no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193C2×Dic3—81no—
194SL(2,𝔽3)#12611yescharacteristic, complemented
195C24C2×S411yescharacteristic
196C2×Q8—31no—
197C2×Q8—31no—
198C2×Q8—31no—
199C2×Q8—32no—
200C2×Q8—41no—
201C2×Q8—61no—
202C2×Q8—61no—
203C2×Q8—61no—
204C2×Q8—61no—
205C2×Q8—61no—
206C2×Q8—122no—
207C2×Q8—121no—
208C2×Q8—121no—
209C2×Q8—121no—
210C2×Q8#23511yescharacteristic
211C2×Q8#23711yescharacteristic
212C2×Q8#23711yescharacteristic
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4C2×S411yescharacteristic
228C22×C4C2×S411yescharacteristic
229C2×C8—61no—
230C2×C8—61no—
231C2×C8—61no—
232C2×C8—61no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—61no—
237C2×C8—61no—
238C2×C8—61no—
239C2×C8—61no—
240C2×C8—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—122no—
254C4⋊C4—122no—
255C4⋊C4—122no—
256C4⋊C4—122no—
257C22⋊C4—121no—
258C22⋊C4—122no—
259C22⋊C4—122no—
260C22⋊C4GL(2,𝔽3)14yes—
261C42—61no—
262C42—61no—
263C42—61no—
264C42—61no—
265C42—122no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—82no—
274C22×C3—82no—
275C22×C3—82no—
276C22×C3—161no—
277C22×C3—161no—
278C22×C3—161no—
279C22×C3—161no—
280C3×C4—82no—
281C3×C4—82no—
282Dic3—82no—
283Dic3—82no—
284Dic3—82no—
285Dic3—82no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23C2×GL(2,𝔽3)12yes—
293C23C22×S411yescharacteristic
294Q8—31no—
295Q8—31no—
296Q8—31no—
297Q8—61no—
298Q8—62no—
299Q8—62no—
300Q8—121no—
301Q8—121no—
302Q8#87911yescharacteristic, complemented
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4C2×GL(2,𝔽3)12yes—
339C2×C4C2×GL(2,𝔽3)12yes—
340C8—121no—
341C8—121no—
342C8—121no—
343C8—121no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—162no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#591211yescharacteristic
360C22#591411yescharacteristic
361C22#591411yescharacteristic
362C22C22×GL(2,𝔽3)11yescharacteristic
363C22#592311yescharacteristic
364C22#592711yescharacteristic
365C22#592711yescharacteristic
366C4—22no—
367C4—22no—
368C4—31no—
369C4—31no—
370C4—31no—
371C4—31no—
372C4—121no—
373C4—121no—
374C4—121no—
375C4—241no—
376C4—241no—
377C4—241no—
378C4—241no—
379C3—41noSylow
380C2—42no—
381C2#8529411yescharacteristic
382C2#8532511yescharacteristic
383C2#8535211yescharacteristic
384C2#8537811yescharacteristic
385C2#8537911yescharacteristic
386C2#8539711yescharacteristic
387C2#8539711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C22⋊C4D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#126
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C4⋊#859C2semidirect
5. #73059C2semidirect
6. #73059C2semidirect
7. C4×#859C2semidirect
194. SL(2,𝔽3)#126semidirect
302. Q8#879semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22⋊C4×SL(2,𝔽3)2
#731482
C4⋊#8592
#730592
C4×#8592
#275223
C2×#8794